But the Premier League side suffered a late scare when skipper Liam Cooper headed against the post in the second minute of stoppage time with Black Cats keeper Costel Pantilimon beaten.
Van Aanholt’s 33rd-minute strike settled the third round tie at the Stadium of Light by the same 1-0 scoreline which separated the teams at Wembley 42 years ago, when the then second division Wearsiders defied the odds to beat the top-flight aristocrats.
Gus Poyet’s side dominated the first half as the visitors attempted to soak up pressure, but the Yorkshiremen responded after the break in front of a travelling contingent of 5,200 among a crowd of 30,302 at the Stadium of Light.
Adryan and debutant Brian Montenegro both tested Pantilimon inside the opening five minutes of the second half, but neither could find the finish to drag their team back into the game, although Cooper went agonisingly close to snatching a replay at the death.
Minds inevitably went back to 1973 and the last and most famous FA Cup meeting between the two clubs during the build-up to the game, and eight of the men who secured that remarkable victory for another generation of Black Cats were presented on the pitch before kick-off.
But unlike that day, it was Leeds who arrived as the overwhelming underdogs and with manager Neil Redfearn making seven changes as he attempted to protect his squad for a Bet Championship campaign which has seen them lose five of their last seven games.
Opposite number Poyet made six changes, with two of those to hand rare starts to Italy international Emanuele Giaccherini and Argentina counterpart Ricky Alvarez after injury, and the pair were to play significant roles as the Barclays Premier League outfit dominated the opening 45 minutes.
Giaccherini was first to catch the eye, forcing keeper Marco Silvestri to save at his near post after tricking his way past full-back Gaetano Berardi with 12 minutes played, and Alvarez’s shot from the resulting corner was deflected on to the top of the crossbar.
Steven Fletcher looked certain to open the scoring eight minutes later when he slid in to meet Van Aanholt’s driven cross, but only succeeded in spooning the ball high over.
The pressure finally told 12 minutes minutes before the break when Fletcher rolled the ball into Van Aanholt’s path and the Dutchman, making his return from a 10-game absence with a dislocated shoulder, smashed an unstoppable shot across Silvestri and in off the far post.
Having already lost Jack Rodwell to injury before the break – he was replaced by Sebastian Larsson – Poyet sent on Connor Wickham for Alvarez at the break, but it was Pantilimon who had to be at his best within seconds when he denied Adryan with his legs after the midfielder had been played in by Antenucci.
The Romanian was called upon once again four minutes later to beat away Montenegro’s shot, and Larsson was relieved to see referee Mike Dean award a free-kick rather than a penalty for his trip on full-back Charlie Taylor on the edge of the box seconds later.
The visitors kept going to the final whistle and very nearly got their reward in added time when Cooper met Murphy’s cross, but was denied by the woodwork.